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Properties

Label 2-3528-72.43-c0-0-2
Degree $2$
Conductor $3528$
Sign $0.642 - 0.766i$
Analytic cond. $1.76070$
Root an. cond. $1.32691$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.5 − 0.866i)2-s + (0.258 + 0.965i)3-s + (−0.499 − 0.866i)4-s + (0.965 + 0.258i)6-s − 0.999·8-s + (−0.866 + 0.499i)9-s + (−0.866 + 1.5i)11-s + (0.707 − 0.707i)12-s + (−0.5 + 0.866i)16-s + 1.93·17-s + i·18-s − 0.517·19-s + (0.866 + 1.5i)22-s + (−0.258 − 0.965i)24-s + (−0.5 + 0.866i)25-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)2-s + (0.258 + 0.965i)3-s + (−0.499 − 0.866i)4-s + (0.965 + 0.258i)6-s − 0.999·8-s + (−0.866 + 0.499i)9-s + (−0.866 + 1.5i)11-s + (0.707 − 0.707i)12-s + (−0.5 + 0.866i)16-s + 1.93·17-s + i·18-s − 0.517·19-s + (0.866 + 1.5i)22-s + (−0.258 − 0.965i)24-s + (−0.5 + 0.866i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.642 - 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.642 - 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3528\)    =    \(2^{3} \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.642 - 0.766i$
Analytic conductor: \(1.76070\)
Root analytic conductor: \(1.32691\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3528} (2059, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3528,\ (\ :0),\ 0.642 - 0.766i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.342984027\)
\(L(\frac12)\) \(\approx\) \(1.342984027\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 + 0.866i)T \)
3 \( 1 + (-0.258 - 0.965i)T \)
7 \( 1 \)
good5 \( 1 + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.866 - 1.5i)T + (-0.5 - 0.866i)T^{2} \)
13 \( 1 + (0.5 - 0.866i)T^{2} \)
17 \( 1 - 1.93T + T^{2} \)
19 \( 1 + 0.517T + T^{2} \)
23 \( 1 + (0.5 - 0.866i)T^{2} \)
29 \( 1 + (0.5 + 0.866i)T^{2} \)
31 \( 1 + (0.5 - 0.866i)T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + (-0.965 - 1.67i)T + (-0.5 + 0.866i)T^{2} \)
43 \( 1 + (0.866 - 1.5i)T + (-0.5 - 0.866i)T^{2} \)
47 \( 1 + (0.5 + 0.866i)T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + (-0.258 - 0.448i)T + (-0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - 0.517T + T^{2} \)
79 \( 1 + (0.5 + 0.866i)T^{2} \)
83 \( 1 + (-0.707 + 1.22i)T + (-0.5 - 0.866i)T^{2} \)
89 \( 1 - 1.41T + T^{2} \)
97 \( 1 + (0.965 - 1.67i)T + (-0.5 - 0.866i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.382459398169363816435816250880, −8.100099369269773315936517471611, −7.66257833859116405182610786086, −6.33231933735621005714239008851, −5.44359065904453569151544148119, −4.91306033273901568603568541319, −4.22305238531004807180170689558, −3.32417106842084341689523718941, −2.64609342594937052948965688691, −1.58311241627985061897308659216, 0.64000086400745112209743176605, 2.33971004068772555019600452656, 3.23035895290954522824569615620, 3.84826288297800173923075199762, 5.30698786805843451942333691332, 5.68538084693639965627384665735, 6.34397313602941390333628641944, 7.24506399373038889181316390121, 7.85637287341378737640365179431, 8.359524904163107643939358599262

Graph of the $Z$-function along the critical line